Subspaces Definition and 334 Threads

Homework Statement Determine whether the set of polynomials of degree 3 form a subspace of P(4) Homework Equations P(4) = c_3 x^3 + c_2 x^2 + c_1 x + c_0 The Attempt at a Solution \alpha P(4_1) = \alpha c_3 x^3 + \alpha c_2 x^2 + \alpha c_1 x + \alpha c_0 This just scales the...

hi, I am confused about vector spaces and subspaces. I've just started a book on linear algebra, and i understood the 1st chapter which delt with gaussian reduction of systems of linear equations, and expressing the solution set as matricies, but the 2nd chapter deals with vectors and I'm...

V = F(R, R), the vector space of all real valued functions f(x) of a real variable x. Which are subspaces of V? (A) {f | f(0) = 0} (B) {f | f(0) = 1} (C) {f | f(0) = f(1)} (D) C^0(R) = {f | f is continous} (E) C^1(R) = {f | f is differentiable and f' is continous}...

Consider 2-by-2 matrices \mathbf{A} =\left( \begin{array}{cc}a & b \\c & d \\\end{array} \right) \in \mathbbm{R}^{2 X 2}. Which of the following are subspaces of \mathbbm{R}^{2 X 2}? (A) {A | c = 0} (B) {A | a + d = 0} (C) {A | ad - bc = 0} (D) {A | b = c} (E) {A | Av = 2v}, where...

Greetings, I'd like to know how one goes about finding a basis for the intersection of two subspaces V and W of a given vector space U. I am aware of the identity V \cap W = (V^{\per} \cup W^{\per})^{\per} (essentially the orthogonal space of the union of orthogonal spaces of V and W), but this...

i've been having some trouble with my linear algebra homework and I am wondering if you guys could give me some insight or tips on these problems: Let v be any vector from V, and let a be any real number such that av=0. Show that either a=0 or v=0. - i was thinking about assuming the...

A and B are two subspaces contained in a finite vector space V and dimA = dimB Can we conclude A=B? In that subspaces A and B are really the same subspace and every element in one is in the other? I think yes because if dimA=dimB then their basis will contain the same number of vectors...

prove that Z(v,T)=Z(u,T) iff g(T)(u)=v, where g(t) is prime compared to a nullify -T of u. (which means f(t) is the minimal polynomial of u, i.e f(T)(u)=0). (i think that when they mean 'is prime compared to' that f(t)=ag(t) for some 'a' scalar). i tried proving this way: suppose, g(T)(u)=v...

How do you find the intersection of subspaces when the subspaces are given by the span of 3 vectors? For example, U is spanned by { X1 , X2 , X3} and V is spanned by { Y1, Y2, Y3}. Thanks in advance.

infinitely many "subspaces" in R3 ? In R3, there are zero, 1, 2, 3 dimensional subspaces. But how can I express them with 'specific' example, using variables x,y,and z?

I'm so lost! 1. W is the set of all vectors in R4 such that x1 + x3 = x2 + x4. Is W a subspace of R4 and Why? How do i get started here? I'm thoroughly confused on this whole idea of vector spaces and such.

Suppose U is a subspace of V. Then U+U = U+{0}=U, right? So the operation of addition of vector spaces does not have unique additive identities. *typo in title

Hello, just wondering if my proof is sufficient. Here is the question from my book: Show that the following sets of elements in R2 form subspaces: (a) The set of all (x,y) such that x = y. ------- So if we call this set W, then we must show the following: (i) 0 \in W (ii) if v,w \in W, then...

Hi, can someone shed some light on the following question. It's been bothering me for a while and I'd like to know where I went wrong. Here is what I can remember of the question. The following is an inner product for polynomials in P_3(degree <= 3): \left\langle {f,g} \right\rangle =...

So the parametric equations of a hypersurface in VN are x^1=acos(u^1) x^2=asin(u^1)cos(u^2) x^3=asin(u^1)sin(u^2)cos(u^3) ... x^(N-1)=asin(u^1)sin(u^2)sin(u^3)...sin(u^(N-2))cos(u^(N-1)) x^N=asin(u^1)sin(u^2)sin(u^3)...sin(u^(N-2))sin(u^(N-1)) where a is a constant. How do I find the...

Hey, I have no problems dealing with vectors in space, R^3. But I am having a lot of trouble with vectors in R^n. One of my basic questions is what is R^n. I mean doesn't the vector space already encompass everything? How do I visualize R^n vectors? Can you reccomend any good online tutorials...

see attached equation thank you!

Hi, I'm wondering how I would decide how many "subspaces of each dimension Z_2^3 has." The answer is: 1 subspace with dim = 0, 7 with dim = 1, 7 with dim = 2, 1 with dim = 3. I'm looking for subsets of Z_2^3 which are closed under addition and scalar multiplication. An arbitrary vector in...

Vector Spaces, Subspaces, Bases etc... :( Hello. I was doing some homework questions out of the textbook and i came across a question which is difficult to understand, could somebody please help me out with it? -- if U and W are subspaces of V, define their intersection U ∩ W as follows...

Hello... I've been doing some home work on Vector Spaces and Vector Subspaces and I need help solving a problem... Can somebody please help me? Consider the differential equation f'' + 5f' + 6f' = 0 Show that the set of all solutions of this equation is a vector subspace of the...

Yet another problem I need to get some starting help on: Show that the set of continuous functions f=f(x) on [a,b] such that \int \limits_a^b f(x) dx=0 is a subspace of C[a,b] Thank you

Hi can someone please help me with the following question. Such questions always trouble me because I don't know where to start and/or cannot continue after starting. Q. Let H and K be subspaces of a vector space V. Prove that the intersection of K and H is a subspace of V. By the way...

I have 2 subspaces U and V of R^3 which U = {(a1, a2, a3) in R^3: a1 = 3(a2) and a3 = -a2} V = {(a1, a2, a3) in R^3: a1 - 4(a2) - a3 = 0} I used the information in U and substituted it into the equation in V and I got 0 = 0. So, does it mean that the intersection of U and V is the whole...

Q: Determine whether U is a subspace of R^3. U = {[0 s t]^T | s and t in R} A: Yes. U = span {[0 1 0]^T, [0 0 1]} Can someone explain to me how the heck they come up with that answer? Seems so random.

How do separation axioms carry over to subspaces? Some are clear -- it's easy to see that if any two points of a space X are separated by neighborhoods, then the same must be true of any subset S of X. But what about the nicer ones? Is it true that if S is a subset of a normal space, that...

Hi everyone- any help would be great! For each integer n>= 2, there exists a vector space V and a linear operator T : L(V ) such that V has exactly n T- invariant subspaces. I think it is true but i do not know how to prove it... awesome thanks!

Having some difficult with general concepts of metric spaces: 1) What is the difference between a subset and a subspace. let's say we have metric space X. and A is a set in that space. Is A necessarily a metric space itself? 2) Why is the metric of X ( d(x,y) for x,y belonging to X )...

I had a question regarding subspaces. Given vectors (a,b,c,d) s.t. \left{\mid}\begin{array}{cc}a&b\\c&d\end{array}\right{\mid}=0 a supspace of \Re^4? Though i kno the answer is yes, but i don't understand like it looks to me that it uses one of the three properties of subspaces; the zero...

Does every linear operator have a nontrivial invariant subspace? My professor mentioned this question in class, but never actually answered it. I am curious if this is true or not and why.

I have to prove or give a counter example to the statement if U1, U2, W are subspaces of V such that V=U1 direct sum W and V=U2 direct sum W, then U2=U1. This is what I did: Let v be an element of V. Then v=v1+v2 for v1 an element of U1 and v2 and element of W and v=v3+v2 for v3 an...

Hey guys, I have a little problem here: given two subspaces U and W both of dimension two of an N dimensional space show in general that if N = 3 the intersection of U and W forms a curve; if N = 4 a finite number of points; and N > 4 they do not in general intersect at all. I can kind...

Hey guys, I have a little problem here: given two subspaces U and W both of dimension two of an N dimensional space show in general that if N = 3 the intersection of U and W forms a curve; if N = 4 a finite number of points; and N > 4 they do not in general intersect at all. I can kind...

I hope someone can help me (guide) in this theorem. How can I show that a "subset W of a vector space V is indeed a subspace of V if and only if given u and v as vectors in W and a and b are said to be scalars, then au + bv is in W."? Can I assume a vector with my desired number of...

I can't solve a problem about subspaces. Help would be great! U and V are subspaces in the vector space R^4[x] given with: U={p(x)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4; a1+a2+a3+a4=0, a1+a2+2a3+2a4=0, a0+a1=a3+a4} V=L{x^3-x^2+x, x^4+1} Find the dimensions and basis for U, U+V and U?V. Is...